0123_0132_0321_1032_1203
Counting sequence:
1, 1, 2, 6, 19, 55, 158, 456, 1319, 3815, 11032, 31900, 92243, 266735, 771306, 2230350, 6449399, 18649427, 53927682, 155940174, 450924959, 1303918761, 3770481322, 10902925724, 31527483945, 91166561073, 263622110548, 762303813502, 2204318533343, 6374125526301, 18431762746840, 53298272296960, 154120138635569, 445662046992879, 1288700243123486, 3726474641115374, 10775673648682093, 31159515028431037, 90102522446544806, 260545279469931998, 753406683973120383, 2178591117099395277, 6299730751623814070, 18216638832069605606, 52676208463755362265, 152321345539997564523, 440460560540906610518, 1273659346326259871178, 3682981578400768736785, 10649906779206235746143, 30795840813038388638394, 89050902608246712280344, 257504359224574222130401, 744613396131011824777129, 2153163975038628416890496, 6226204265855621101418376, 18004025708011052301172747, 52061404967441754247242463, 150543546823417205931111050, 435319782559607393926305780, 1258793997394157472517514087, 3639996139294660807071568085, 10525607781343184115448383352, 30436411173815195432007437398, 88011556613724282040469822549, 254498930683220601714367685903, 735922738001008306572721804176, 2128033602549858724612219857636, 6153535934875714409885601572453, 17793894070297957934074580084723, 51453777069944061646130570959590, 148786497452671931542047866116404, 430239004497984428246447819873165, 1244102147443168468182984013618095, 3597512399134314980661987733356292, 10402759523020869615885989224624658, 30081176570744332845427873387669149, 86984341210410814820075475834135783, 251528579609077178915028484697871112, 727333511753811403187619277214942686, 2103196535925735491819797314601610417, 6081715742842415115121167648252274497, 17586214956586111790303058195360473788, 50853241022196646906582173693644698552, 147049955254478248344865722706244035079, 425217526074801244529706082331110483149, 1229581771502564967654229252397161868786, 3555524502405924419630893322958261582950, 10281345072121968365182741713491591571865, 29730088042008524499435487264115476396429, 85969114817693257325478924551077716084770, 248592896600060705959021904093394610966532, 718844533540431108305928056175005959124535, 2078649351885116568642065884281381536707827, 6010733790812634085118682511532969611349060, 17380959742561525934960655020954047624016558, 50259714052600168332547978832627820857719839, 145333680882046553975595167575444276455756309, 420254655182062484758199812031519709146615672, 1215230868235800805855794298128195486740405850, 3514026661937547924191731351619673922449406669
Generating function in Maple syntax:
(x^4-x^3-2*x+1)/(x^7-2*x^3+x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{4}-x^{3}-2 x +1}{x^{7}-2 x^{3}+x^{2}-3 x +1}
Generating function in sympy syntax:
(x**4 - x**3 - 2*x + 1)/(x**7 - 2*x**3 + x**2 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^7-2*x^3+x^2-3*x+1)*F(x)-x^4+x^3+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+2 x -1 = 0
Explicit closed form in Maple syntax:
646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+5)+646059/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+5)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+4)-179107/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+4)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+3)-2840885/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+3)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+2)-202135/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+2)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n+1)+462107/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n+1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n-1)+664782/16286641*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n-1)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 1)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 2)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 3)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 4)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 5)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 6)^(-n)+4954371/32573282*RootOf(_Z^7-2*_Z^3+_Z^2-3*_Z+1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{32573282}+\frac{646059 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{32573282}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{16286641}-\frac{179107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{16286641}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{32573282}-\frac{2840885 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{32573282}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{16286641}-\frac{202135 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{16286641}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{32573282}+\frac{462107 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{32573282}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{16286641}+\frac{664782 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{16286641}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{32573282}+\frac{4954371 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{3}+\textit{\_Z}^{2}-3 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{32573282}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 158
a(n) = 2*a(n+4)-a(n+5)+3*a(n+6)-a(n+7), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 158
a \! \left(n \right) = 2 a \! \left(n +4\right)-a \! \left(n +5\right)+3 a \! \left(n +6\right)-a \! \left(n +7\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16221/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[60,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[31,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[12,x]+F[30,x]
F[30,x] = F[18,x]
F[31,x] = F[32,x]+F[37,x]
F[32,x] = F[15,x]+F[21,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[2,x]*F[4,x]
F[37,x] = F[15,x]+F[38,x]+F[54,x]+F[58,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[16,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[30,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = 2*F[15,x]+F[38,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[35,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[32,x]*F[4,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[48,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = 2*F[15,x]+F[47,x]+F[53,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[32,x]+F[52,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[34,x]+F[62,x]
F[62,x] = F[32,x]+F[57,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{60}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{30}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{35}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{52}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{34}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{57}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_60(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_31(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_12(x) + F_30(x))
Eq(F_30(x), F_18(x))
Eq(F_31(x), F_32(x) + F_37(x))
Eq(F_32(x), F_15(x) + F_21(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_2(x)*F_4(x))
Eq(F_37(x), F_15(x) + F_38(x) + F_54(x) + F_58(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_16(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_30(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), 2*F_15(x) + F_38(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_35(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_32(x)*F_4(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_48(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), 2*F_15(x) + F_47(x) + F_53(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_32(x) + F_52(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_34(x) + F_62(x))
Eq(F_62(x), F_32(x) + F_57(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 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